The main difference between this work and that of other researchers is that we include a periodic modulation in the model parameters, investigating the dynamics of these models in addition to their variability/sustainability from the point of view of informa- tion theory. An alternative approach in this direction is to include a stochastic noise rather than a periodic modulation in components in the above models.
The above observations may be useful to all people concerned with controlling the growth of this terrifying disease cancer and that it would be useful to study the above systems with inclusion of a time delay in one or more species as, according to observa- tions by Marchuk (see [6,59]), involving a time delay in the immune system modelling is a significant factor in terms of its influence on the cancer model, with and without immunotherapy.
The models investigated in this thesis present only a few of the possible modifications of models for eradication of cancer cells, and suggestions for future work could include constructing the growth of the cancer or the interaction for the cancer and immune cells by using a non-exponential function or conducting a similar investigation on models containing medicine resistance and more kinds of specific immune cells.
from our suggested models and strengthen our findings to enable their future inclusion in the treatment of cancer sufferers.
A dimensionless form for the
predator-prey model
As discussed earlier, the predator-prey model behaviour has been studied by many researchers in its deterministic conditions as well as stochastic variations. Here, re- placing the predator-prey model’s exponential growing of the prey population by a logistic growth equation with a carrying capacity N= K1 yields the model:
dx
dt = ax(1 − Nx) − bxy, (A.1a)
dy
dt = −cy + exy. (A.1b)
Our purpose is to simplify the above model and work with a dimensionless version. It has two species in addition to five parameters, which means that there are lots of choices for the dimensionless parameters. By dividing the first equation (A.1a) by a and the second equation (A.1b) by c as follows:
dx dat = x(1 − Nx) − b a xy, dy dct = −y + e c yx. (A.2)
We rescale the parameters in Eqs. (A.2) according to:
u= b
a x, v=
e
c y and T = a t,
where T is a time constant. As a result, the following model has been emerged from the evolution of Eqs. (A.2) with the latter quantities
d dT a b u= a b u(1 − N a b u) − c euv, (A.3a) 1 c ad dT c ev= − c ev+ a buv. (A.3b)
For simplicity, we multiply Eq. (A.3a) by ba and Eq. (A.3b) by ea to derive the follow- ing equations: du dT = u(1 − a bNu) − bc aeuv, dv dT = −c a v+ e buv. (A.4)
Furthermore, lets consider the following quantities in Eqs. (A.4) in order to get a dimensionless form. N = b a, α = bc ae, β = c a and γ = e b.
Then, the predator-prey model can be displayed as follows after a dimensionless pro- cedure (fewer parameters) is applied:
du
dT = u(1 − u) − αuv,
dv
dT = −βv + γuv.
(A.5)
As a result, we obtain a model with less parameters which is easier to investigate and examine its dynamical properties such as the equilibrium point (u∗,v∗) that can be
A dimensionless form for the three
species model
We expand the predator-prey model by including a third species which represents the healthy cells population as we can see from the following equations:
dx
dt = ax(1 − N1x) − bxy − dxz, (B.1a)
dy dt = −cy + eyx g2+ x − f yx + F sin2(ωt), (B.1b) dz dt = gz(1 − N2z) − hzx. (B.1c)
By dividing the first equation (B.1a) by a, the second equation (B.1b) by c and the third equation (B.1c) by g, we obtain the following equations:
dx dat = x(1 − N1x) − b axy − d axz, dy dct = −y + eyx c(g2+ x) − f cyx + F c sin 2 (ωt), dz dgt = z(1 − N2z) − h gzx. (B.2)
follows: T = at, u = e cx, v= b ay and r = h gz.
As a result, we obtain the following model after including all the above dimensionless quantities and more simplification:
c e du d T = c eu(1 − N1 c eu) − c euv − cdg aehur, a2 cb dv d T = − a bv+ a b vu g2+ ceu − a f bevu + F c sin 2 (ω aT ), a h dr d T = g hr(1 − N2 g hr) − c eru. (B.3)
Furthermore, lets consider the following quantities in Eqs. (B.3) in order to get a form with less parameters.
N1= e c, N2 = h g, Ω = ω a and α = c e.
The three species model in its final form can be display as follows, as a result of applying a dimensionless and simplification procedure.
du d T = u(1 − u) − uv − a13ur, dv d T = a21 " vu g2+ αu − v # − a22vu + a23sin2(ΩT ), dr d T = a31r(1 − r) − a32ru. (B.4)
Fisher information construction for
the logistic model
By following the procedure presented by several researchers, we calculate Fisher in- formation index for the logistic model. We display the logistic model expressed in Eq. (2.3) as follows:
dx
dt = [B + N0sin(ωt)] x (1 − x
K). (C.1)
We fix the constant growth B to be B= 0. Since Fisher information quantity as in Eq. (2.14) is: FT = A T Z T 0 ˙u2 u4 dt, (C.2)
where ˙u = dudt and u = dxdt, A is a normalization constant and T is the total time duration. We may now differentiate the function u with respect to x as follows:
˙u = ∂u ∂x
dx
FT = A T Z T 0 (N0 sin(ωt) ( ˙x − 2x ˙xK ))+ (N0ω cos(ωt) (x − x 2 K)) u2 dt, (C.4)
We employ the formula in (C.4) to produce the figures in subsection (2.4.4) for different parameter values. Similar form for Fisher information is calculated for the Gompertz equation.
Fisher information construction for
the predator-prey model
Fisher information can be calculate based on the background theory explained by sev- eral researchers. In the following, we present the procedure of obtaining Fisher infor- mation for the predator-prey model. We display the predator-prey model expressed in Eqs.(4.1) by the following set of equations:
dx
dt = ax − bxy = f1(x, y), dy
dt = −cy + exy = f2(x, y).
(D.1)
We expand Fisher information quantity in Eq. (2.14) as follows:
FT = A T Z T 0 ˙s2 s4 dt, (D.2)
where s = p ˙x2+ ˙y2, A is a normalization constant and T is the total time duration. In consideration of the above, we obtain ˙s= dsdt as follows:
˙s= ∂s ∂x dx dt + ∂s ∂y dy dt, (D.3)
∂x ∂y
we may now differentiate the function s with respect to cancer population x as follows: ∂s
∂x =
˙x(a − by)+ ey˙y
s , (D.4)
and the next derivative is for s with respect to y as follows: ∂s
∂v =
−bx ˙x+ ˙y(−c + ex)
s , (D.5)
substituting the results of Eqs. (D.4) and (D.5) together in Eq. (D.3) and finally in- cludes the outcome results in Eq. (D.2), we obtain the following Fisher information quantity: FT = A T Z T 0
˙x2(a − by)+ ˙x˙y(ey − bx) + ˙y2(−c+ ex) 2
˙x2+ ˙y2
3 dt, (D.6)
The latter formula (D.6) is used to produce the figures in subsection (4.2.4), and similar calculations can be follow in order to construct Fisher information equation in the other sections in Chapter 4 with small differences.
Fisher information construction for
the three species model
We calculate Fisher information by an integral equation, and its computation is subject to a group of ordinary differential equations (ODEs). Here, we show how to calculate Fisher information for the three species model.
We display the dimensionless form for the three species model which is expressed in (5.2) by the following set of equations:
du dT = u (1 − u) − uv − a13ur, dv dT = a21[ uv g2+ αu − v] − a22uv + a23sin2(Ω1T ), dr dT = a31r(1 − r) − a32ur. (E.1)
Since Fisher information quantity is:
FT = A T Z T 0 ˙s2 s4 dt, (E.2) where s = √
˙u2+ ˙v2+ ˙r2, A is a normalization constant and T is the total time duration. In consideration of the above Fisher information (Eq. E.2), we obtain ˙s= dsdt
∂t = ∂u dt ∂v dt ∂r dt or the derivative of s in equations (E.3) could be written as:
˙s= ∂s ∂u ˙u+ ∂s ∂v ˙v+ ∂s ∂r ˙r,
we may now apply the calculus of a simple differentiation for the function s with respect to cancer population u as follows:
∂s ∂u = ˙u(1 − 2u − v − a13r)+ ˙v( a21v(g2+αu)−a21αuv (g2+αu)2 − a22v) − a32r˙r s , (E.4)
the next derivative is for s with respect to v as follows: ∂s
∂v = a21u˙v
g2+αu− u ˙u − a21˙v − a22u˙v
s , (E.5)
and the evolution equation of the derivative of s with respect to r can be as follows: ∂s
∂r =
a31˙r − a13u˙u − 2a31r˙r − a32u˙r
s , (E.6)
by substituting the results of Eqs. (E.4), (E.5) and (E.6) together in the Eq. (E.3) and finally includes the outcome results in Eq. (E.2), we obtain the following Fisher information quantity: FT = A T Z T 0
˙u2a+ ˙v2b+ ˙r2c+ ˙u˙vd − ˙u˙re2
˙u2+ ˙v2+ ˙r2
where: a= 1 − 2u − v − a13r, b= a21u g2+ αu − a21− a22u, c= a31− 2a31r − a32u, d = a21v(g2+ αu) − a21αuv (g2+ αu)2 − a22v − u, e= a32r+ a13u.
As a result, we obtain Fisher information index which it has been used to produce Fisher information figures in Section (5.2), similar quantity can be compute for dif- ferent dynamical systems by following the same procedure in Appendixes C, D, & E.
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